Courant Institute of Mathematical Sciences - …qidi/publications/tracer_twolayer.pdfPassive scalar turbulence concerns about the advection and diffusion of a tracer ﬁeld pas-sively

PREDICTING EXTREME EVENTS FOR PASSIVE SCALAR TURBULENCE INTWO-LAYER BAROCLINIC FLOWS THROUGH REDUCED-ORDER

STOCHASTIC MODELS

DI QI∗ AND ANDREW J. MAJDA†

Abstract. The capability of using imperfect stochastic reduced-order models to capture crucial passive tracerstatistics such as tracer energy spectrum, tracer intermittency, and eddy diffusivity is investigated. The passive scalarfield is advected by a two-layer baroclinic turbulent flow which can generate various representative regimes in atmo-sphere and ocean. Much simpler and more tractable linear Gaussian stochastic models are proposed to approximatethe complex and high-dimensional advection flow equations. The imperfect model prediction skill is improved througha judicious calibration of the model errors using leading order statistics of the background advection flow, while noadditional prior information about the passive tracer field is required. A systematic framework of correcting modelerrors with empirical information theory is introduced, and optimal model parameters under this unbiased informationmeasure can be achieved in a training phase before the prediction. It is demonstrated that crucial principal statisticalquantities like the tracer spectrum and fat-tails in the tracer probability density functions in the most important largescales can be captured efficiently with accuracy using the reduced-order tracer model in various dynamical regimes ofthe flow field with distinct statistical structures. The skillful linear Gaussian stochastic modeling algorithm developedhere should also be useful for other applications such as accurate forecast of mean responses and efficient algorithmsfor state estimation or data assimilation.

Key words. passive tracer turbulence, intermittency, low-order stochastic model

AMS subject classifications. 76F25, 86A32, 62B10

1. IntroductionPassive scalar turbulence concerns about the advection and diffusion of a tracer field pas-

sively transported by a turbulent dynamical flow. The problem has many applications in atmo-sphere, ocean, and climate change sciences, as well as many other branches of environmentalscience, such as the behavior of anthropogenic and natural tracers and the spread of contami-nants or hazardous plumes [8, 9, 24, 19, 11]. Important and representative characteristics in thepassive tracer field that can be observed in atmospheric observations, laboratory experiments,and numerical simulations include the extreme events represented by non-Gaussian exponentialtails in probability density functions of the tracer field [10, 24, 23, 3, 4, 5]. Another remark-able property of these systems is their complex multiscale structures in both space and time,thus intermittent energy back cascades along the spectrum add important small scale feedbacksto tracer large scale variability. Major difficulties in accurate prediction about the large-scaletracer statistics are due to the inadequate numerical resolution for small scale feedbacks andincomplete physical understanding about the energy transport mechanism of the system. Oneway to address such complications is to build instructive simplified models which maintain theskill in generating the key statistical features of the realistic passive tracer field to understandthe major mechanism in the passive tracer turbulence, so that the essential dynamical structurescan be identified [19, 8, 17, 22].

The scalar field T (x,t) describes the concentration of the passive tracer immersed in thetwo-dimensional fluid on x=(x,y) which is carried with the local fluid velocity v but does notitself significantly influence the dynamics of the fluid. We consider this issue in the contextof the evolution of the scalar field through the joint effect of turbulent advection and diffusion

∗Department of Mathematics and Center for Atmosphere and Ocean Science, Courant Institute of MathematicalSciences, New York University, New York, NY 10012, email([email protected])

†Department of Mathematics and Center for Atmosphere and Ocean Science, Courant Institute of MathematicalSciences, New York University, New York, NY 10012, email([email protected])

1

2 Predicting Extreme Events for Passive Scalar Turbulence through Reduced-Order Stochastic Models

[29, 31, 19]

∂T∂ t

+v·∇T =DT =−dT T +κ∆T. (1.1)

In general we assume the dissipation as a linear damping and diffusion, DT =−dT T +κ∆T ,where dT has a stronger effect on the large-scale modes while κ applies to the small-scale modesfor the diffusion and unresolved effects together. The statistical description of tracer transportin turbulent flows is a prevalent concern in atmospheric and oceanic fluid dynamics when theadvection flow v becomes turbulent [31, 28]. Generally the statistical solution for (1.1) cannotbe obtained in explicit form because of the nonlinear advection term from v·∇T . However withcertain simplifications about a prescribed tracer mean gradient and fluctuations around the meangradient, there exists well developed statistical-dynamical understanding for the above problemwith much simpler master equations for the advection flow with realistic physical characteristics[19, 22]. The mechanism of the existence and behavior of intermittency is investigated under arigorous mathematical framework. Fat-tailed probability density functions and random spikesin the time series of the tracer can also be generated by numerical simulations using the idealizedmodel (1.1) with a wide range of scales in the advection flow field [17, 3, 34].

In this paper, the the advection velocity field v(x,t) is modeled by the two-layer quasi-geostrophic (QG) system [29, 33]. Despite its relatively simple structure, the two-layer QGmodel can generate a wide variety of dynamical regimes that capture many essential featuresof both the atmosphere and ocean [29, 32, 36]. Typically there are anisotropic, intermittent,and inhomogeneous structures, and strong non-Gaussian statistics may appear in the flow fieldespecially among the dominant large scales. For example, in the atmosphere regime it wouldexhibit a representative flow shifting between a cross-sweep zonal jet in the east-west directionand the transverse Rossby waves in the north-south direction. Theories and parameterizationsfor the anisotropic transport of a tracer in this two-dimensional anisotropic flow with jets aredeveloped in [2, 31]. A multi-scale stochastic superparameterization strategy has also beendeveloped for the system and has been successfully applied in predicting the passive tracerequilibrium spectra and in multi-scale filtering [15]. The complexity and large computationalexpense in resolving the highly turbulent true advection flow equations require the introductionof simpler and more tractable imperfect models which still maintain the ability in capturing thekey intermittent features in the tracer field. Notice that the central quantity of interest here is thetracer statistics rather than the background flow field. Thus it is advantageous to develop simplerreduced-order stochastic models that can capture the most important features of the backgroundadvection flow v, and check the performance and model errors in the resulting scalar tracer Tdue to the imperfect modeling of flow solutions.

In turbulence literature, additional dissipation is introduced for coarse resolution numericalmodeling of the turbulent diffusion of passive tracers, which is usually called eddy diffusion[19, 25, 29]. Stochastically excited linear system has been designed for modeling the nonlineareddy interactions in QG turbulence [7, 6], but the procedure there often reports limited skill incapturing the mean response of the true system accurately [18]. A class of physics-constrainedmulti-level nonlinear regression models was developed which involves both memory effectsin time as well as physics-constrained energy conserving nonlinear interactions [18, 12] withmathematical rigor. In this paper, we address the reduced-order modeling problem for obtainingcrucial principal statistics of the passive scalar field immersed in a turbulent high-dimensionalfluid field with inhomogeneous statistics. The primary goal is to obtain a mathematically un-ambiguous reduced-order stochastic modeling strategy with high skill for the scientific inves-tigation of complex turbulent diffusion problems arising in many applications for which noexplicit solution is available. We begin with simple model approximation considering linearGaussian dynamics in the advection flow equations, then calibrate the model errors and over-

Di Qi and Andrew J. Majda 3

come the possible barriers due to using this crude approximation of the flow field v. Only loworder equilibrium statistics are required in the model calibration as a correction to the nonlinearsmall scale feedbacks, thus high computational cost can be avoided. The challenge here is todevelop efficient reduced-order stochastic models for the flow velocity field that keep the skillin capturing the leading order moments, as well as the higher order statistics that characterizeintermittency in the principal directions in the tracer system.

In designing the reduced-order models, we propose imperfect model parameters to quan-tify the unresolved small scale feedbacks [28] in the stochastic advection flow approximation,and then use the imperfect flow solution to predict the intermittent structures in the turbulentpassive tracer field from reduced-order tracer equations of (1.1). Therefore a systematic pro-cedure in calibrating the imperfect model errors due to the crude approximations is necessary.In a training phase before the prediction, an information-theoretic framework [16, 26, 13] isproposed to tune the imperfect model parameters in addition to the equilibrium consistency inthe leading order moments, so that the model predicted stationary process can possess the leastbiased estimation in energy and autocorrelation functions compared with the truth. The majorstatistical quantities to check for the reduced-order model prediction skill will include the mean,variance of the leading dominant tracer spectral modes that determine the major structure in thetwo-dimensional turbulent tracer field, as well as the fat-tailed tracer PDFs. The reduced-ordermodel is then tested in three typical dynamical regimes representing the ocean and atmospherewith distinct statistics. Despite strong non-Gaussianity and anisotropic structures in the originalturbulent advection flow, the linear stochastic model displays promising predicting skill for thetracer statistics and intermittency in large scale leading modes with high accuracy and effectivereduction in computational cost.

The remainder of this paper is organized as follows. Section 2 introduces the spectralformulations for both the background flow equations and the scalar passive tracer equationsadvected by the flow field. Then Section 3 describes the calibration strategies for the linearGaussian stochastic advection flow model so that consistency in first two leading order momentsand autocorrelation functions can be guaranteed. The reduced-order tracer equations can beapplied using the solution from the optimized flow equations. The skill of the reduced-ordermodels is then tested under the two-layer quasi-geostrophic system in Section 4. A summarydiscussion about the results is shown in the final section of this paper.


2. Spectral dynamical formulation for the turbulent diffusion models with mean gra-dient

First we consider the general formulation for a scalar tracer T (x,t) which is advected bya velocity field v(x,t). The turbulent advection flow is described from the solution of the two-layer QG equation [29, 33]

∂q j

∂ t+v j ·∇q j+

(β +k2

dU j) ∂ψ j

∂x=−δ2 jr∆ψ j−ν∆sq j,

q j =∆ψ j+k2

d2(ψ3− j−ψ j

), v j =(U j,0)+v′j.

(2.1)

Above the subindex j=1,2 is used to represent the upper and lower layer of the two-layerflow model. The two dimensional incompressible velocity field v j is decomposed into a zonalmean cross sweep, (U j,0), and a fluctuating shear flow v′j =∇⊥ψ j =(−∂yψ j,∂xψ j). Next weintroduce the passive tracer equation. Assume the existence of a background mean gradient forthe tracer field varying in x and y directions and a tracer fluctuation component

Tj (x,t)=α·x+T ′j (x,t), (2.2)

where α=(αx,αy) is the horizontal mean gradient in x and y directions. In the atmosphereanalogy a tracer gas would include a north-south mean gradient and fluctuations restricted to anarrow latitude band. The existence of a background mean gradient for the tracer field has beenshown responsible for many important phenomena [1, 3, 10].

With the tracer decomposition into mean gradient and fluctuations around the mean in (2.2),the fluctuation parts of the tracer in upper and lower layers satisfy the following dynamics forthe full fluctuation passive tracer model of tracer field T ′j (x,t), j=1,2

∂T ′jε−1∂ t

+v′j (x,t)·∇T ′j +U j∂T ′j∂x

=−(αxu j+αyv j)(x,t)−dT T ′j +κ∆T ′j . (2.3)

Above v′j =(u j,v j) is the fluctuating advection flow field from the solution of equations (2.1)together with a zonal mean flow U j. We further introduce a scale separation in the tracer equa-tion (2.3) with order ε , while keeping the solution of the advection flow (2.1) unchanged. Thedifference in time scale in the tracer is through a different time scale, t=ε−1t, in the tracer timelike in various previous works [19, 17, 22]. As ε <1, the velocity field is varying at a fastertime scale than the passive tracer process, while on the other hand with ε >1 the tracer evolvesin a more rapid rate than the advection field. A long time rescaling limit with explicit analytictracer solutions is derived in [22] and numerical simulations for varying values of ε among awide range are investigated in [3] under a much simpler linear model. In general, different in-termittent features will be generated from near Gaussian statistics to distributions with fat tailsas the scale separation parameter value changes [3, 19].

In this section, we first derive the explicit equations for the two-layer advection flow fieldas well as the passive tracer field in spectral space. They can offer a detailed illustration aboutthe energy mechanism especially from the nonlinear effect between the tracer and underlyingflow field. Through a thorough understanding about the true model mechanism, the basic ideasabout constructing reduced-order models can be revealed. The reduced-order model is thenconstructed with a careful calibration according to the true model nonlinear interactions.

2.1. Exact passive tracer and advection flow dynamical equations in spectral spaceGiven periodic boundary condition in both the two-layer flow and the tracer field, we for-

mulate the flow and tracer fields with Galerkin truncation to finite number of Fourier modes.


Spatial Fourier decomposition in flow potential vorticity q j and passive tracer disturbance T ′jcan be written in the expansion under modes exp(ik·x) as

q j =∑k

q j,keik·x, T ′j =∑k

Tj,keik·x.

By projecting the tracer and flow equations (2.3) and (2.1) to each Fourier spectral mode,equations for the spectral coefficients in each wavenumber of the two-layer tracer field ~Tk=(T1,k,T2,k

)T , and two-layer advection flow field ~qk=(q1,k,q2,k)T , form the set of ODEs in the

spectral domain as

d~Tk

dt+ε−1 ∑

m+n=k

(Akm~qm~Tn+Akn~qn~Tm

)=−ε−1(γT,k+iωT,k)~Tk+ε−1Gk~qk, (2.4a)

d~qk

dt+ ∑

m+n=k(Akm~qm~qn+Akn~qn~qm)=−

(γq,k+iωq,k

)~qk, (2.4b)

where ‘’ is used to denote the pointwise produce, ab=(aibi). The potential vorticities~qk andstream functions ~ψk in two layers are related by the transform matrix Hk

~qk=Hk~ψk=−[|k|2+ k2

d2 − k2

d2

− k2d2 |k|2+ k2

d2

]~ψk,

through the relation q j =∇2ψ j+k2

d2

(ψ3− j−ψ j

)in (2.1). Note the core nonlinearity are dom-

inated by the quadratic exchange of energy among triad modes on the left hand sides of (2.4)with the same explicit form, Akm= 1

2 (kxmy−kymx)H−1m , for both the tracer and flow equations.

On the right hand side of the tracer dynamics (2.4a), shear flow from the velocity field α·vintroduces the operator Gk applied on the flow vorticity as an external forcing, and the tracerdamping γT,k and phase shifting operators ωT,k are due to dissipation and the advection in large-scale zonal mean flow respectively, that is,

Gk=−iα·k⊥H−1k =ΓkH−1

k , γT,k=dT +κ |k|2 , ωT,k=kx~U .

In the advection flow field dynamics (2.4b), the operators get the explicit forms

γq,k=(0,1)T r|k|2H−1k +ν |k|2s , ωq,k=kx

(~U+

(β +k2

d~U)

H−1k

),

as linear dissipation and dispersion effects. γq,k is due to the Ekman friction only applied onthe bottom layer and the hyperviscosity, and ωq,k is from the rotational β -effect as well as thebackground zonal mean flow advection from the original equation (2.1) applied on the vorticitymodes.

The above spectral models (2.4) for tracer advected by two-layer baroclinic turbulent flowcan be compared with the simpler master models discussed in detail in [19, 17]

dTk+ε−1ikUt Tkdt=−ε−1(γT,k+iωT,k)

Tkdt−ε−1α vkdt,

dvk+ikUt vkdt=−(γv,k+iωv,k

)vkdt+σv,kdWk,t ,

dUt =−γUUtdt+σU dWt .

(2.5)

The simpler formulation in (2.5) models the nonlinear interaction through the zonal jet Ut froma mean zero Ornstein-Uhlenbeck process in the third equation for both the tracer and flow dy-namics, and similar linear dissipation and dispersion terms are applied on the right hand sides


of the tracer and flow equations. Explicit solutions for the mean, variance, and cross-covariancebetween tracer and flow modes can be derived [17] due to this simple structure, and intermit-tency in such system can be predicted through the random resonance between modes of theturbulent velocity and the passive tracer [22]. Motivated from the theoretical understandingachieved from the simplified system (2.5), the system (2.4) advected by the two-layer QG flowcontains more realistic features with inherent internal instability and strong non-Gaussian statis-tics. Thus we are able to investigate the more complicated triad nonlinear interactions betweenthe flow and tracer modes in different scales, and the effects of non-Gaussianity in the advectionflow field to the final tracer distributions instead of the simple large-scale Gaussian advectionin the system (2.5). More importantly, ideas about reduced-order modeling strategies can beproposed by comparing the more complicated system with strong non-Gaussianity (2.4) and thesimpler master equations (2.5).

2.2. Reduced-order stochastic formulations for the passive tracer and turbulent flowfield

The advection terms in the tracer and flow equations (2.4) involve interactions betweenmodes of different scales along the entire spectrum in a large dimensional phase space, thususually high computational cost is required in achieving accurate statistical results from directnumerical simulations. In general intermittency in tracer field is dominated by the variabilityin largest scales, thus we will concentrate on the large-scale modes with wavenumber |k|≤MN, where M is the number of resolved modes and N is the full dimensionality of thesystem. Usually we could choose M much smaller than N that only covers the essential mostenergetic directions in the flow system. Below we first propose the simple strategy with linearcorrections to approximate the advection flow field in the leading modes. Then the calibrationand improvement of the imperfect models due to model errors from this approximation will bediscussed in Section 3.

2.2.1. Consistent linear Gaussian model in the advection flowIn approximating the advection flow, we begin with the simple Gaussian approximation by

replacing the quadratic interactions (v·∇q)k in the flow equations by additional linear dampingand random Gaussian noise. From the exact solutions from the simpler system (2.5), it hasbeen shown that important non-Gaussian and intermittent structures in the tracer field can begenerated from a linear Gaussian advection flow. It is useful to check the optimal predictionskill for the nonlinear tracer statistics using only linear Gaussian approximation in the flow field.

Therefore the reduced-order advection flow equations are proposed as

d~qM,k=−(γq,k+iωq,k

)~qM,kdt−DM

q,k~qM,kdt+ΣMq,kd~Wq,k, 1≤|k|≤M. (2.6)

vM =∇⊥~ψM, ~qM,k=Hk~ψM,k,

with only Gaussian statistics generated. Only the first M large-scale modes, 1≤|k|≤M, areresolved in the reduced-order formulation in (2.6).

(γq,k,ωq,k

)are the linear dissipation and

dispersion operators as in (2.4b) that are easy to model. On the other hand, the expensivebut crucial part in the original equations (2.4b) is the nonlinear interactions Akm~qm~qn wheremodes with different scales are coupled. Especially the nonlinear term represents the energytransfers between different modes so that the unstable modes due to internal instability can bebalanced. Thus additional damping and noise

(DM

q ,ΣMq)

are introduced to correct model errorsdue to the neglected nonlinear interactions in the flow equations.

One of the simplest and most direct way to estimate the undetermined coefficients(DM

q ,ΣMq)

is through the mean stochastic model (MSM) discussed in [18, 12]. The param-eter values are estimated from the characteristic quantities of total energy and decorrelationtime. Despite the simplicity in the linear autoregressive models, reasonably skillful prediction


in uncertainty quantification as well as filtering can be reached under this strategy for turbulentsystems [18, 12]. However MSM still suffers several shortcomings when strong nonlinearityand slow mixing take place in the system [26]. Thus more detailed calibration of the imperfectmodel parameters are required especially when non-Gaussianity becomes crucial in the system.The optimization strategy for the undetermined imperfect model parameters will be discussedin detail next in Section 3 using detailed calibration from equilibrium statistics.

2.2.2. Direct stochastic modeling in the passive tracer equationsIn the tracer dynamics, we would not introduce additional model calibrations of the tracer

field statistics in case of over fitting of data. The idea here is to improve the reduced-ordermodel prediction skill by optimizing the background advection flow field, thus the reduced-order passive tracer equations can be modeled through a direct truncation

d~TM,k+(

vM ·∇~TM

)k

dt=Γk~ψM,kdt−(γT,k+iωT,k)~TM,kdt, 1≤|k|≤M. (2.7)

vM = ∑|k|≤M1

ik⊥~ψM,keik·x, M1≤M.

Consistent with the advection flow model (2.6), only the first leading modes |k|≤MN areresolved in the tracer approximation model.

Again the major difficulty in modeling the tracer dynamics is from the accurate approxima-tion of the tracer advection Akm~qm~Tn in (2.4a). Exact modeling about this nonlinear interactionterm requires the flow mode solution~qM,k along the entire spectrum 0< |k|≤N, while only thefirst M leading modes are available through the reduced-order model. One crude approximationidea could be to replace the nonlinear advection in the tracer field, v(x,t)·∇~T (x,t), with lineardamping and noise in a similar fashion as the flow approximation model (2.6). However, asdiscussed in previous works [19, 22], the nonlinear advection in the tracer equation is crucialin the generation of many important statistical features including the intermittency. Thus, theinclusion of nonlinear effects from the flow solution is essential, at least for the large scalemodes. On the left hand side of the equation (2.7), the nonlinear advection vM ·∇~TM is mod-eled explicitly, but only the first M1≤M largest scale flow modes in the model velocity solutionvM are used to calculate the imperfect model tracer advection. This nonlinear advection is es-sential in generating the accurate spectra in tracer statistics, while it is also not expensive tocalculate since only leading modes are involved. The idea for this approximation is through theassumption that the dominant features in tracer statistics (such as intermittency and equilibriumspectrum in large scales) are due to the leading advection flow modes with largest energy. Thecontribution from the leading dominant modes in tracer statistics will be further illustrated inSection 4.1 through explicit model simulations.


3. Imperfect model calibration using equilibrium statistics and information theoryIn this section, we discuss the strategies for the optimal prediction models through calibra-

tion of low-order statistics in the advection flow system. In the stochastic reduced-order modelproposed in (2.6) for the advection flow equation, efficient linear Gaussian dynamics are used,while large model errors are introduced due to the approximation in the turbulent flow field.Strategies for finding optimal imperfect model parameters that minimize the model error areproposed here according to the corresponding statistical equations of the flow equation. In thismodeling process, we only consider the calibration of the advection flow field, and the flowequation solution is applied to the tracer equations for predicting tracer statistics directly.

3.1. Model calibration in the advection flow field

3.1.1. Statistical equations for the advection flow fluctuationIn constructing reduced-order flow models, we need to make sure that the imperfect model

calibration parameters(DM

q ,ΣMq)

can properly reflect the true nonlinear energy mechanism fromthe true system. The consistent imperfect model then can be proposed by consulting the modelstatistical dynamics. Therefore it is useful to investigate the statistical equations for the second-order moments from the fluctuation equations of (2.4b). The dynamics for the covariance matrixRq

k=〈~qk~q∗k〉 of flow vorticity can be derived as a 2×2 blocked system for each wavenumber[28, 21]

dRqk

dt+Lk(q)Rq

k+RqkL

∗k (q)+Qq

F =(L q

k +Dqk)

Rqk+Rq

k(L q

k +Dqk)∗, |k|≤N. (3.1)

The linear operators (L q,Dq) represent the skew-symmetric dispersion and dissipation effectsfrom the right hand side of (2.4b). The additional operator Lk(q) represents the interactionswith a non-zero statistical mean state, where internal instability occurs with positive growthrate. Most importantly, the nonlinear interactions between different spectral modes introducethe additional nonlinear flux term Qq

F indicating higher-order interactions, that is,

QqF (~qk)=

12 ∑

m+n=k〈(Akm~qm~qn+Akn~qn~qm)~q∗k〉. (3.2)

Therefore the small and large scale modes are linked through third-order moments 〈~qm~qn~q∗k〉in (3.2) between the triad modes m+n=k. The nonlinear flux Qq

F plays the central role inthe energy mechanism that balances the unstable directions due to internal instability fromthe linear operators. Thorough discussions about the energy mechanism and construction ofstatistical closure models for (3.1) are detailed in [28, 21, 27] for more generalized systems.

Here our focus is on the low-order stochastic realization in (2.6) of the statistical closuremodel of (3.1), thus solving the statistical equation (3.1) directly is not favorable consideringits complexity. Still the nonlinear flux Qq

F (3.2) corresponds to the unresolved nonlinear effectsin the stochastic model in (2.6). Thus it is useful to exploit the nonlinear flux Qq

F so that theimperfect model parameters

(DM

q ,ΣMq)

in (2.6) can be proposed according to the true modelenergy transfer mechanism. Especially in statistical equilibrium as t→∞ the nonlinear fluxescan be calculated easily from the localized lower-order moments

QqF,eq=

(L q

k +Dqk−Lk

(qeq))

Rqk,eq+Rq

k,eq

(L q

k +Dqk−Lk

(qeq))∗

. (3.3)

Above in (3.3) only steady state statistics about the first two moments(qeq,R

qeq)

are required,thus we do not really need to calculate the entire spectrum to get the higher order feedbacks inthe dominant low wavenumber modes.


3.1.2. Imperfect model correction from equilibrium statistics and additional correc-tion terms

As illustrated in above sections, the expensive nonlinear flux QqF in the statistical equa-

tions (3.1) is replaced by linear damping and noise with parameters(DM

q ,ΣMq)

in the stochasticmodels (2.6) representing the unresolved interactions. In fact, we can decompose the matrixQq

F =Qq,+F +Qq,−

F by singular value decomposition into positive-definite and negative-definitecomponents. The positive definite part Qq,+

F illustrates the additional energy that injected intothis mode from other scales, while the negative definite part Qq,−

F shows the extraction of energythrough nonlinear transfer to other scales.

In the first step of constructing the stochastic damping and noise parameters, the idea is tomake use of the equilibrium statistics of the nonlinear flux Qq

F,eq which can be achieved easilyfrom the lower-order equilibrium statistics as in (3.3). In adopting the true equilibrium statisticsfrom Qq

F,eq, the true model energy transfer mechanism is respected and least artificial effectis introduced into the imperfect approximation model. Considering all these aspects, the firstproposal for the linear damping and Gaussian random noise correction can be introduced as

Deqq,k=−

12

Qq,−F,eq,k

(Rq

k,eq

)−1, Σeq

q,k=(

Qq,+F,eq,k

)1/2. (3.4)

The additional damping is from the negative definite equilibrium flux Qq,−F,eq; and the positive

definite equilibrium flux Qq,+F,eq acts as additional noise to the system. The above additional

damping and noise (3.4) offer a desirable quantification for the minimum amount of correctionsto stabilize the system with consistent equilibrium statistics for the mean and variance. Thisis the same idea applied to the statistical modified quasi-linear Gaussian closures developed in[30].

Still the above estimation of parameters (3.4) may not be optimal for the reduced-orderGaussian model considering that: i) it only guarantees marginal stability in the unstable modesfor equilibrium; and more importantly ii) the time mixing scale in each mode (represented bythe autocorrelation functions) may still lack the accuracy in the approximation using only equi-librium information. The nonlinear energy transferring mechanism may change with large de-viation from the equilibrium case when intermittent fluctuations are present. The shortcomingsfor purely using the approximation (3.4) only from equilibrium statistics can be observed fromthe various numerical simulations [26]. As a further correction, we propose additional terms ontop of (3.4) with a simple constant damping for all the spectral modes and an additional noiseaccordingly to make sure consistency in energy

QaddM,k=−Dadd

M RM,k+(

ΣaddM,k

)2, Dadd

M =diagdM+iωM,dM−iωM. (3.5)

Specifically for the two-layer system, we introduce the additional damping parameter dM withsame rate in the upper and lower layer. The imaginary part of the parameter ωM introducesadditional dispersion effect for correcting the time decay scale among the two layer modes.This further correction term in (3.5) is aimed to offer stabilizing effects in the marginal stableequilibrium form (3.4), and to offer further corrections in modeling the autocorrelation functionthat is important for the mixing rate in each spectral mode.

Combining the ideas in (3.4) and (3.5), We propose the additional damping and noise cor-rections for the reduced-order flow vorticity model (2.6) in the following form

DMq,k=−

12

Qq,−F,eq,k

(Rq

k,eq

)−1+Dadd

M , ΣMq,k=

(Qq,+

F,eq,k+(

ΣaddM,k

)2)1/2

. (3.6)


Above the parameters(

DMq,k,Σ

Mq,k

)are 2×2 matrices applied on the two-layer spectral mode~qk.

Comparing with the exact true system (2.4b), the reduced-order approximation is equivalent toreplacing the nonlinear interaction terms with the judiciously calibrated damping and noisein consideration with both the equilibrium energy transfer mechanism and further sensitivitycorrection. Next the task left in the Gaussian linear flow model is a systematic strategy toestimate the optimal imperfect model parameters in (3.5) in a training phase making sure thati) statistical equilibrium consistency for the mean and variance can be maintained; and ii) thetime mixing scale represented by the autocorrelation function in each mode can be modeledcorrectly with the truth.

3.2. Imperfect stochastic model consistency in equilibrium statistics and autocorrela-tion functions

From the above construction of reduced-order stochastic models, we still have the unde-termined model parameters

(Dadd

M ,ΣaddM)

in the linear damping and noise corrections in (3.6).These additional parameters can offer the freedom to control the imperfect model consistencyin the leading order moments, and agreement in autocorrelation functions. Now we describe themodel calibrations for consistency in equilibrium moments, and model optimization using au-tocorrelation functions under a systematic information-theoretic framework. This information-theoretic framework based on the relative entropy offers an unbiased and invariant measure formodel distributions [14, 13], and has been utilized to systematically improve model fidelity andsensitivity and to make an empirical link between model fidelity and forecasting skill [16].

3.2.1. Climate consistency of the stochastic models for the first two momentsFirst we calculate the constraints for the imperfect model parameters so that the resolved

modes can converge to the same equilibrium statistics in the first and second order moments instatistical steady state. The linear parts form the same linear dissipation and dispersion operatorsas in (3.1), thus we only need to consider the imperfect model approximation of the nonlinearflux compared with the truth in QM and Qq

F . The contribution from the additional damping DMq

in (3.6) adds the effective outflow of energy in each mode as

Q−M,k=DMq,kRM,k+RM,kDM∗

q,k =−12

Q−F,eq,kR−1k,eqRM,k+Dadd

M RM,k+c.c.

The additional noise ΣMq in (3.6) acts as the inflow of energy in each mode as

Q+M,k=

(ΣM

q,k)2=Qq,+

F,eq,k+(

ΣaddM,k

)2.

In the statistical steady state, we would like to make sure the imperfect model statistics convergeto the truth as

RqM→Rq

eq, as t→∞.

From the negative and positive definite correction terms above in Q−M and Q+M , the first compo-

nent acts as the nonlinear flux approximation from equilibrium statistics, thus it will convergeto the truth QF,eq in the equilibrium. Therefore we get the necessary condition for the additionaldamping and noise to make sure climate consistency so that

(Σadd

M,k

)2=Dadd

M Rk,eq+Rk,eqDadd∗M . (3.7)

In the two-layer case, DaddM is defined in (3.5) for the 2×2 uniform damping in each spectral

mode, and the 2×2 matrix ΣaddM,k adds additional noise into each spectral mode accordingly.


3.2.2. Training phase for optimal autocorrelation functions with information theory

By choosing parameters according to (3.7), the climate consistency for the imperfectreduced-order models in the unperturbed equilibrium is guaranteed. Still the imperfect stochas-tic model may not be optimized due to the incorrect modeling about time scales and mixing ratein each mode (See Figure 4.5 in Section 4.2.2 as an example). On the other hand, we still havethe additional model parameters (dM,ωM) in (3.5) to control the performance in each mode. Thetime mixing scale in the stochastic process for~qk(t) can be characterized by the autocorrelationfunctions defined as

Rk(t)≡⟨(~qk(τ)−〈~qk〉)(~qk(τ+t)−〈~qk〉)∗

⟩, (3.8)

that forms a 2×2 matrix for each spectral mode. It measures the lagged-in-time covariance⟨~q′(t)~q′(0)∗

⟩in statistical stationary state as ‘memory’ in remembering the previous state in

each mode, thus characterizes the mixing rate. Next we follow the general strategy introducedin [26] for measuring autocorrelation functions with the help of information theory.

The problem is about to find a proper metric to measure the errors in the imperfect modelautocorrelation functions in the training phase in a balanced way. A preferable measure to offeran unbiased metric for the imperfect model probability distribution πM from the truth π is touse the relative entropy [14, 16, 13]

P (π,πM)=

ˆπ log

ππM

. (3.9)

However one additional difficulty in tuning the autocorrelation functions under the informationmetric is that the functions given in (3.8) is not actually a distribution function to measure inthe relative entropy (3.9) above. The autocorrelation function R (t) may oscillate with negativevalues, thus it becomes improper to directly substitute R (t) into the formula (3.9) by replacingthe distribution function π to measure the distance. The problem can be solved by insteadconsidering the spectral representation of the random process ~q(t), from the theory of spectralrepresentation of stationary random fields [35]. It is proved a positive-definite matrix E (λ )>0can be constructed so that the spectral representations of the autocorrelation R (t) and stationaryprocess~q become

R (t)=ˆ ∞

−∞eiλ tdF (λ ), ~q(t)=(q1,q2)

T =

ˆ ∞

−∞eiλ t~Z(dλ ), (3.10)

where the spectral random measure ~Z(dλ )=(Z1(dλ ),Z2(dλ )

)T has independent incrementsand energy spectrum measured by E (λ ) or dF (λ )

dF (λ )=E (λ )dλ =E∣∣∣~Z(dλ )~Z∗(dλ )

∣∣∣.

Here specifically for the application in the two layer flow system, we view the spectral densityand distribution functions E and dF as 2×2 matrices (see Appendix A for more details about thespectral expansion). In this way, we can optimize the autocorrelation functions R (t) includingupper and lower layer covariances at the same time since these two modes are always closelycoupled together.

Applying the theory for spectral representation of stationary processes, we find the one-to-one correspondence between the autocorrelation function R (t) and positive-definite energyspectra E (λ ). Back to the comparisons of the true and model random fields ~q,~qM constrainedwithin first two moments, R (t) for the true process~q can be achieved through the data from true


model simulations in (2.4b), while the imperfect model autocorrelation RM (t) can be solvedexplicitly through the Gaussian linear process in (2.6). E (λ ) and EM (λ ) then are the Fouriertransforms of the autocorrelation matrices R (t),RM (t) according to (3.10). Therefore we canconstruct the spectral measures for two Gaussian random fields as a product of increment inde-pendent normal distributions about the frequency λ

pG(x;λ )=∏N (0,E (λ )dλ ), pMG (x;λ )=∏N (0,EM (λ )dλ ).

The normalized relative entropy (3.9) between the two processes then can be defined under thespectral densities

P(

pG,pMG)=P (E (λ ),EM (λ )),

ˆ ∞

−∞P(

pG(x;λ ),pMG (x;λ )

)dλ . (3.11)

See [26] and Appendix A for a more detailed derivation about the above formula. We abuse thenotation above using the spectra E (λ ) to denote density functions. Since E and EM are positivedefinite 2×2 matrices for the spectral random variables, it is well-defined of the last part ofthe above formula (3.11) using the information distance formula (3.9). Through measuring theinformation distance in the spectral coefficients ~Z(dλ ), we get the lack of information in theautocorrelation function R (t) from the model. Furthermore, we have shown in [26] that theerror in autocorrelation functions ‖R (t)−RM (t)‖ of two stationary random processes ~q and~qM is bounded by the information distance of their energy spectra P (E,EM).

To summarize, we can seek a spectral representation of the autocorrelation functions like

P (E (λ ),EM (λ ))=ˆ

D(

E (λ )EM (λ )−1)

dλ , (3.12)

where D (x)≡−logdetx+trx−2 is the Gaussian relative entropy with a zero mean state [13].And since we only concentrate on the leading order statistics (that is, mean and variance),thus this representation is enough. We can find the optimal model parameter θ∗=(dM,ωM) byminimizing the information metric defined in (3.12)

P (E (λ ),EM (λ ,θ∗))=minθ

P (E (λ ),EM (λ ,θ)). (3.13)

The same strategy has been tested successfully for the simpler L-96 system in [26].

3.3. Summary about the numerical algorithm for the reduced-order stochastic mod-els

With the optimal reduced-order stochastic approximation for the background advectionflow equations constructed in (3.6) in which consistent leading order statistics and autocorrela-tion functions are guaranteed, the prediction about tracer statistics can be carried out by apply-ing the reduced-order tracer equations as in (2.7). As a summary, here we describe the generalreduced-order stochastic modeling procedure for predicting passive tracer statistics advected bya two-layer turbulent flow field.ALGORITHM 3.1. (Reduced-order stochastic model for passive scalar turbulence)

Determine the low-dimensional subspace spanned by orthonormal basis ek covering theregime with largest variability in the flow spectrum. The stochastic dynamical equations foradvection flow and passive tracer field (2.6) and (2.7) can be set up by Galerkin projecting the


original equations to the resolved subspace for wavenumbers 1≤|k|≤M,

d~TM,k=−(

vM ·∇~TM

)k

dt+Γk~ψM,kdt−(γT,k+iωT,k)~TM,kdt,

d~qM,k=−(γq,k+iωq,k

)~qM,kdt−DM

q,k~qM,kdt+ΣMq,kd~Wq,k,

vM =∇⊥~ψM = ∑|k|≤M1

ik⊥~ψM,keik·x, ~qM,k=Hk~ψM,k.

The flow vorticity equation for~qM needs to be calibrated in a training phase for optimal modelparameters, and then the solution from the flow equation is used for the prediction of tracerstatistics in ~TM in the most energetic leading modes.

• Calibration in advection flow equations: in the vorticity equations about ~qM , addi-tional linear damping and Gaussian noise

(DM

q ,ΣMq)

are introduced according to theflow equilibrium statistics and further corrections in (3.6). Equilibrium consistencyis guaranteed through the constraint in (3.7), and the optimal model parameter isachieved through tuning the imperfect model autocorrelation function with the truthunder information metric using (3.12) and (3.13).

• Prediction in passive tracer equations: the advection flow solution vM is calculatedfrom reduced-order model with the optimized parameters. Using the advection flowsolution, the truncated tracer model is applied to get the statistics in passive tracerfield. Tracer energy spectrum can be achieved by averaging the time-series of thetracer solution, and the tracer intermittency can be checked by comparing the tails intracer probability density functions.


Table 4.1: Model parameters representing test regimes in ocean and atmosphere. The oceanregime gets a larger deformation frequency F than the atmosphere regime. N is the modelresolution, β is the rotation parameter, F = f 2

0 L2/4π2g′H=k2d/4 is inversely proportional to

the deformation radius, U is the background mean shear flow, r is the Ekman drag in the bottomlayer, and the hyperviscosity is measured by the operator−ν∇2s. The last three columns displaythe parameters for the tracer equations for linear damping dT , tracer diffusivity κ , and meangradient α along y.

regime N β F U r ν s dT κ αocean, high lat. 128 2 40 0.1 0.1 1×10−13 4 0.5 0.001 1

atmosphere, high lat. 128 1 4 0.2 0.2 1×10−13 4 0.1 0.001 1atmosphere, mid lat. 128 2 4 0.2 0.1 1×10−13 4 0.1 0.001 1

4. Numerical experiments for passive tracer statistics in various regimesIn this section, we test the reduced-order models described in the previous sections

among different dynamical regimes of the two-layer QG flow with distinct statistical fea-tures. In numerical simulations, the true statistics are calculated by a pseudo-spectra code with128×128×2 grid points in total. We use this relatively small truncation since it already sufficesto generate strong large-scale intermittency. The zonal mean flow ~U =(U,−U) is taken as thesame strength with opposite directions in the two layers. In the tracer simulations, for simplicitywe always consider the mean gradient along y direction, that is to assume, T =T ′+αy. Thisassumption is representative in many previous investigations [22, 17, 26]. We may change thescale separation parameter ε in (2.4a) to generate different tracer structures. In most of the testcase here, we use ε−1=5 where intermittency is prominent. Parameter values for both oceanand atmosphere regimes are following [13, 20] and are shown in Table 4.1. In the reduced-order models, we only compute the modes |k|≤M=10 in largest scales, compared with thetrue system resolution N=128.

4.1. Statistics in the true system with full resolutionIn the first place, we display the true model solutions with full resolution as an illustration

of the flow structures. Using the two-layer baroclinic model in (2.1), many typical dynami-cal regimes with distinct statistical features can be generated by varying the parameter values.Here we focus on three representative regimes as shown in Table 4.1. In the high latituderegimes, the statistics are relatively homogeneous with a competition between the zonal modesand meridional modes. The typical structure is the exchange between the blocked regime withstrong meridional heat transfer and unblocked regime with strong zonal jet (see Figure 4.3and [29, 28]). The mid latitude atmosphere regime forms another interesting dynamical regimewith representative anisotropic zonal jets (see Figure 4.1 and [29]). Thus stronger non-Gaussianstatistics can be expected in this regime.

As a typical illustration about the advection flow field in the three test regimes in Table4.1, we compare the zonal averaged flow time-series, u=−

fflψy(x,y)dx, in Figure 4.1. In the

high latitude ocean regime, there is obvious competition between the unblocked zonal flowand blocked meridional heat flux (see Figure 4.3 for the snapshots of stream functions in thesetwo regimes). In the high latitude atmosphere regime, similar structure can be observed withstronger turbulent mixing. In contrast, the mid latitude atmosphere regime generates a mean-dering zonal jet, which displays the highly anisotropic structure in this flow field. These threedynamical regimes get different statistics and share representative features with the typical flowfields that are observed in real applications.


Fig. 4.1: Time-series of the zonal averaged flow in the three test regimes with parameters inTable 4.1. The evolutions of the high latitude ocean (first row), the high latitude atmosphere(second row), and the mid latitude atmosphere (third row) are compared. The x-axis is time andy-axis is meridional coordinate.

4.1.1. Statistics in the advection flow fieldIn the statistics of the advection flow field, most importantly we consider the first two

leading moments of the mean and variance in each mode (and they are also important in con-structing the reduced-order model for the flow dynamics); but it is still useful to check the higherorder statistics about how large the steady state statistics divert from the Gaussian distributions.Usually to measure the higher order moments, two characterizing statistical quantities are theskewness and flatness of the variables of interest, that is,

skewness=

⟨(u−u)3

⟩

⟨(u−u)2

⟩3/2 , flatness=

⟨(u−u)4

⟩

⟨(u−u)2

⟩2 . (4.1)

The skewness is used to measure the symmetry of the distributions of the random variables;flatness (kurtosis) indicates how the peak and tails of a distribution differ from the normaldistribution. For a normal distribution, flatness gets the standard value of 3. A distribution with aflatness larger than 3 indicates that the distribution has tails heavier than the normal distribution(with same variance) and sharper peak. Likewise, a distribution with a flatness smaller than 3indicates that the distribution has tails lighter than a normal distribution and flatter peak in themiddle. One additional characterizing quantity is the autocorrelation function, R (u), which canbe used to measure the time mixing scale of the process. The integrated correlation time of theprocess can be defined in two ways

Mabscorr=

ˆ|R (t)|dt, Mcorr=

∣∣∣∣ˆ

R (t)dt∣∣∣∣. (4.2)

Both Mabscorr and Mcorr are useful in measuring the mixing rate of the state variables for both the

advection flow and the passive tracer modes [1, 19] especially when the autocorrelation functionis highly oscillatory.

The equilibrium steady state statistics in the first four leading modes,ψ(1,0),ψ(0,1),ψ(1,1)ψ(−1,1), of the flow stream functions are listed in Table 4.2 for thethree test regimes. In the high latitude regimes, the first two modes (1,0) and (0,1) are dominantwith much larger energy than the other modes, representing the Rossby waves and the zonal


Table 4.2: Equilibrium steady state statistics for the advection flow field stream functions ψkin three test regimes. The (absolute) mean, variance, skewness, and flatness are comparedin the first four most energetic modes. The original Mcorr (in parenthesis) and absolute Mabs

corrdecorrelation time are shown in the last column, which illustrate the mixing scale for eachmode in the advection flow.

lower layer mean variance skewness flatness correlation timeψ(1,0) 0.0261 14.0134 -0.0136 2.4459 3.1190 (0.4673)ψ(0,1) 0.1593 13.3364 0.0369 2.2103 6.2486 (6.1076)ψ(1,1) 0.0050 0.4332 -0.0349 4.6093 1.3864 (0.2823)

ψ(−1,1) 0.0099 0.4379 0.0310 4.7116 1.4175 (0.2990)high-latitude ocean regime F =40,β =2 (lower layer)

lower layer mean variance skewness flatness correlation timeψ(1,0) 0.0046 0.5917 -0.0073 2.3438 4.1925 (1.1283)ψ(0,1) 0.0215 0.5898 -0.0265 2.1646 3.5128 (2.8489)ψ(1,1) 0.0018 0.0490 -0.0350 3.2605 1.7763 (1.0116)

ψ(−1,1) 0.0023 0.0500 -0.0421 3.1349 1.7144 (0.9326)high-latitude atmosphere regime F =4,β =1 (lower layer)

lower layer mean variance skewness flatness correlation timeψ(1,0) 0.0029 0.0111 0.0055 4.1219 17.7396 (0.7574)ψ(0,1) 0.0002 0.1397 -0.0813 1.7995 97.6285 (73.672)ψ(1,1) 0.0003 0.0176 -0.0557 3.6148 16.6485 (1.2648)

ψ(−1,1) 0.0005 0.0180 0.0082 3.5290 15.4606 (1.0893)mid-latitude atmosphere regime F =4,β =2 (lower layer)

jets respectively. This can also be seen in Figure 4.2 with the energy spectra. For the variancesin upper and lower layers of the flow, ocean regime has relatively the same amount of energyin both layers, and atmosphere regime accumulates more energy in the upper layer. In the midlatitude atmosphere regime, instead the zonal mode (0,1) gets much larger energy representingthe meandering jet structure.

In higher order statistics, first note the skewness are all small within numerical errors forall the modes. This shows the symmetry in the distributions in the leading modes of interest.In high latitude regimes, for the fourth order moments for flatness, the first two modes (1,0)and (0,1) get values smaller than 3, representing sub-Gaussian distributions; while the next twomodes (1,1) and (-1,1) have values larger than 3, representing fat tails. In mid latitude case,the zonal mode (0,1) becomes strongly sub-Gaussian while all the other modes are stronglysuper-Gaussian with fat tails. These higher order statistics can be further observed in Figure4.4 in the marginal PDFs. The correlation time Mcorr and Mabs

corr can be used to measure themixing rate in each mode. Much longer mixing time can be observed also in the two leadingmodes. Another important feature to observe is the difference between Mcorr and Mabs

corr. Modeψ(0,1) gets relatively similar values between Mcorr and Mabs

corr, showing a slowly decaying mode(representing zonal flow); while mode ψ(1,0) has much difference in Mcorr and Mabs

corr, meaningstrong oscillation (representing meridional transport mode).

4.1.2. Intermittency in the passive tracer fieldHere we display the intermittent structure in the passive tracer field briefly in the true model.

As a typical example we use the high latitude ocean regime with representative blocked and un-blocked regimes as shown in Figure 4.3. Correspondingly we can observe the time-series of


100

101

102

wavenumber

10-10

10-5

100

energy in velocity field |v|2, high lat. ocean

upper

lower

100

101

102

wavenumber

10-10

10-5

100

energy in velocity field |v|2, high lat. atm

upper

lower

100

101

102

wavenumber

10-10

10-5

100

energy in velocity field |v|2, low lat. atm

upper

lower

(a) energy spectra in advection flow field

100

101

102

wavenumber

10-5

100

energy in tracer field |T|2, high lat. ocean

upper

lower

100

101

102

wavenumber

10-10

10-5

100

energy in tracer field |T|2, high lat. atm

upper

lower

100

101

102

wavenumber

10-15

10-10

10-5

100

energy in tracer field |T|2, mid lat. atm

upper

lower

(b) energy spectra in passive tracer field

Fig. 4.2: The energy spectra in the three typical test regimes. Both the energy spectra in flowvelocity field, |k|2

⟨|ψk|2

⟩, and the tracer field,

⟨∣∣Tk∣∣2⟩

, are shown in radial average.

the meridional heat flux defined as H f =ffl

vτ =ffl

ψxτ , with (ψ,τ) the barotropic and baroclinicmode. In the unblocked state with strong zonal flow, the meridional heat flux is suppressed;while in the blocked state strong meridional heat flux can be observed showing a strong merid-ional transport of the heat. The source of intermittency in passive scalar tracer is describedqualitatively in [3] under an elementary model. The streamline topology for the flow fieldchanges from unblocked behavior in the x direction (that is, parallel to the imposed tracer fieldmean scalar gradient αy along y), to very rapid heat transport in the y direction in the blockedstate. The open streamlines in the meridional transport (like the left panel in first row in Figure4.3), along with the y mean gradient, lead to large convective transport and large deformationsof the isocontours for the passive scalar field, which promotes strong mixing by diffusion. Onthe other hand, in the regime with strong zonal flow along x direction (like the right panel infirst row in Figure 4.3), weak distortion of the scalar isocontours is caused by the almost en-tirely zonal convection of the flow field, and hence little opportunity for strong tracer mixingby diffusivity. Confirmed in the time-series of the scalar passive tracer trajectory in Figure 4.3,strong tracer intermittent bursts are always corresponding to the strong meridional heat trans-port in blocked regime leading to high mixing rate in tracer modes; the quite regime with littlemeridional heat flux in the unblocked regime also shows little intermittency of the tracer fieldin small amplitude.

The above on and off intermittent mechanism detailed in [3, 19] explains the turbulent mix-ing and intermittency via streamlines blocking and opening through simple elementary modelwith no instability and turbulence, while similar phenomena are also observed in the numericalsimulations through much more complex two-layer baroclinic model in turbulence here. Eventhough simple and qualitative, the intuitive reasoning hints that the first two most energeticleading modes (0,1) and (1,0) which represent the competitive unblocked and blocked statesmight be crucial for modeling the intermittent structures in the tracer field. This also verifiesthe validity of only using small number of leading modes to model the tracer advection in (2.7)


barotropic flow field, blocked regime

-3 -2 -1 0 1 2 3

x

-3

-2

-1

0

1

2

3

y

(a) blocked regime

barotropic flow field, unblocked regime

-3 -2 -1 0 1 2 3

x

-3

-2

-1

0

1

2

3

y

(b) unblocked regime

200 400 600 800 1000 1200 1400 1600 1800 2000

0

1

2time-series of flow heat flux vτ

200 400 600 800 1000 1200 1400 1600 1800 2000

time

-5

0

5time series of tracer T, upper layer

(c) time-series of heat flux and scalar tracer

Fig. 4.3: Illustration about tracer intermittency with heat flux. The first row is the snapshots ofthe flow field with blocked and unblocked regime. The following parts show typical time-seriesfor both the heat flux and scalar passive tracer.

for the reduced-order tracer model.

4.2. Calibration in the advection flow fieldNow we use the reduced-order stochastic model in (2.6) and (2.7) to predict scalar passive

tracer statistics with only calibration of the background turbulent advection flow field. Onlythe leading modes |k|≤10 are resolved for reducing model computational cost. As the modelcalibration strategy described in Section 3.1, we need to confirm equilibrium consistency inthe energy spectra and time scale consistency in autocorrelation functions for the flow vorticitysolutions.

4.2.1. Consistency in energy spectraAs described in the modeling strategy in the previous section, the linear Gaussian flow ap-

proximation equations in (2.6) should recover the true statistical energy in equilibrium in thefirst place. Choosing the parameters according to the constraint (3.7), naturally the same statis-tics can be reached since the true equilibrium first two moments are used to help construct thereduced model approximation for additional damping and noise corrections. To further con-firm this convergence and make sure the validity of the numerical schemes, Figure 4.4 displaysthe equilibrium PDFs from the first 4 leading modes, ψ(1,0),ψ(0,1),ψ(1,1)ψ(−1,1), of flow streamfunctions in truth and reduced-order model predictions. Consistency in variances is confirmedin these marginal distributions for the reduced-order model predictions overlapping the Gaus-sian fit of the truth with the same variance. Besides, consistent with the numbers in Table 4.2,the true leading modes all display some degree of non-Gaussian statistics with sub-Gaussian orsuper-Gaussian structures in the tails. Especially for the first two leading modes, that is, (1,0)for the Rossby waves and (0,1) for the large-scale zonal flow, strongly non-Gaussian features


-10 -5 0 5 10

10-2

mode (1,0)Gaussian fit

truth

red. model

-10 -5 0 5 10

10-2

100

mode (0,1)

-2 -1 0 1 2

100

mode (1,1)

-2 -1 0 1 2

100

mode (-1,1)

(a) high lat. ocean

-2 -1 0 1 2

10-2

100

mode (1,0)

-2 -1 0 1 210

-2

10-1

100

mode (0,1)

-1 -0.5 0 0.5 1

100

mode (1,1)

-1 -0.5 0 0.5 1

100

mode (-1,1)

(b) high lat. atmosphere

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (1,0)

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

100

mode (0,1)

-0.4 -0.2 0 0.2 0.4

100

mode (1,1)

-0.4 -0.2 0 0.2 0.4

100

mode (-1,1)

(c) mid lat. atmosphere

Fig. 4.4: The marginal PDFs of the first four leading modes in stream functions in all threetest regimes. The truth is shown in blue and red is the reduced model result. The Gaussiandistributions with the same equilibrium variance are in dashed black lines. Since we use theequilibrium statistics to correct the linear Gaussian approximation models, exact recovery aboutthe Gaussian fit can be achieved through this reduced-order model.

can be observed in all three regimes. On the other hand, the reduced-order model only capturesthe Gaussian statistics since only linear Gaussian model is used.

4.2.2. Consistency in autocorrelationsStill the climate consistency in the leading order moments is not sufficient for representing

the low-order imperfect approximations in an optimal way. It is observed that large model errorcould still exist in the autocorrelation functions of each mode. As illustrated in Section 4.1.2 andFigure 4.3, the tracer intermittency is related to the competition between the zonal mode (0,1)and meridional mode (1,0). Thus the correct modeling about the time mixing scales in thesemodes is directly linked with the occurrence of intermittency, then with the accurate predictionof the fat-tails in tracer distributions.

The autocorrelation functions of the first four leading modes in the flow stream functionsare plotted in Figure 4.5. Again we observe that the zonal mode (0,1) has a long decayingtime representing the persistent zonal jet especially in mid latitude regime; while the Rossbymode (1,0) is highly oscillating responsible for the intermittent heat transport. Comparing thereduced-order model results with the optimal parameters from (3.13) and the non-optimizedmodel with no additional corrections dM =0,σM =0, it can be seen that in all three test regimes,the tuning process in the training phase can effective improve the accuracy in the predictedautocorrelation functions, while large error may still exist for the non-optimal case without ad-ditional correction. Important time scales in these crucial modes will be missed if no additionalmodel calibration is considered. As we can see in the predictions for tracer statistics in the nextsection, this is also important for the reduced model skill for tracer results.

4.3. Predictions of turbulent tracer statistics in reduced-order modelsIn this section, we test the prediction skill of the reduced-order tracer models in (2.7) using

the solution from the optimized flow equations as achieved above. As we have discussed, thenonlinear advection in the tracer equation vM ·∇TM is important for the final tracer statisticalstructure, while this part is expensive to calculate explicitly since it is nonlocal requiring the


0 5 10 15 20-1

0

1mode (1,0)

0 10 20 30 40 50-1

0

1mode (0,1)

0 2 4 6 8 10-1

0

1mode (1,1)

0 2 4 6 8 10time

-1

0

1mode (-1,1)

(a) high lat. ocean

0 10 20 30 40 50-1

0

1mode (1,0)

truthred. model (optimized)red. model (non-optimized)

0 20 40 60 80 100-1

0

1mode (0,1)

0 5 10 15 20-1

0

1mode (1,1)

0 5 10 15 20time

-1

0

1mode (-1,1)

(b) high lat. atmosphere

0 20 40 60 80 100-1

0

1mode (1,0)

0 50 100 150 200 250 300-1

0

1mode (0,1)

0 20 40 60 80 100-1

0

1mode (1,1)

0 20 40 60 80 100time

-1

0

1mode (-1,1)

(c) mid lat. atmosphere

Fig. 4.5: Autocorrelation functions in the first four most energetic modes in the flow field streamfunctions in all three test regimes. The truth in dashed blacked lines are compared with the op-timized reduced-order model estimations in red lines. The optimal parameters in the reduced-order model is achieved by minimizing the information metric between the autocorrelationswith the truth. For comparison, we also give the non-optimized results without additional cor-rections (dM,σM)=0 in dotted-dashed blue lines.

modes in different scales. So the strategy is to consider only principal modes with largestvariance in vM in calculating the nonlinear term and check the imperfect model prediction skillas in (2.7). Besides, we still have the scale separation parameter ε and linear damping dT tocontrol the tracer structures in the tracer model. This enables us to investigate the reduced modelskill with different statistics by changing these parameter values.

4.3.1. Prediction in tracer equilibrium spectra and autocorrelation functionsIn the reduced-order tracer equation (2.7), we would not like to introduce further calibra-

tions of the tracer field. Thus in the first place, we should check the imperfect model skill inrecovering the energy spectra and autocorrelation functions in the resolved tracer modes. Asfrom the original tracer model (2.4a), the nonlinear advection in the tracer dynamics is non-local, while the reduced-order model only uses the leading modes in the flow field solutionvM =∑vM,keik·x,|k|≤M1, to calculate the nonlinear part. By changing the size of truncatedmodes M1, we can check the contribution of modes in different scales in tracer advection. InFigure 4.6-4.8, the equilibrium spectra in tracer modes and cross-covariance between tracermodes and flow stream functions are shown. To clearly compare the two-dimensional modesalong one axis, the tracer modes are ordered all together with descending energy accordingto the advection flow modes. In the reduced-order model only the leading modes |k|≤10 arecomputed explicitly in the equations. Furthermore, we consider two different truncation sizesM1=10 and M1=2 in calculating the nonlinear advection in vM . Thus with M1=2, only thefirst two dominant modes (1,0) and (0,1) are used in the nonlinear advection velocity, vM ·∇TM .In general, the energy structure in the largest scales can be captured with desirable accuracy,while large errors appear with smaller size of advection modes. The cross-covariances betweenthe tracer mode and stream function can also be captured through the reduced-order formula-tion. Comparing in more details in the first few dominant modes, using only two modes M1=2leads to larger errors due to the inaccurate modeling about the unresolved small scale feed-backs to the largest scales. Especially in the mid latitude regime with persistent zonal jet, using


100

101

102

100

variance spectra <Tk

2>, upper layer truth

M1 = 10

M1 = 2

100

101

102

100


2>, lower layer

100

101

102

wavenumber

100

variance <Tk1

Tk2

> between two layers

2 4 6 810

-1

100

2 4 6 8

100

2 4 6 8

100

(a) tracer energy spectra

100

101

102

100

cross-covariance spectra <Tkψ

k>, upper layer

100

101

102

100


k>, lower layer

100

101

102

wavenumber

100

cross-covariance spectra <Tk1ψ

k2> between two layers

2 4 6 8

100

2 4 6 8

100

2 4 6 8

100

(b) tracer-flow covariance

Fig. 4.6: Tracer variance spectra predicted with the reduced-order model in high latitude oceanregime with tracer parameters

(ε−1,dT

)=(5,0.5). The modes are ordered in descending order

of the corresponding advection flow energy. The truth is shown in black, and we considerdifferent number of resolved modes M1=10 and M1=2 in the reduced model advection.

only two modes M1=2 becomes insufficient with large errors. This shows the importance ofsmall scale contributions in this regime. Overall, using the total M1=10 resolved modes in theadvection, the leading mode energy can be approximated with good agreement with the truth.

Next we check the reduced model prediction of the autocorrelation functions in the lead-ing tracer modes. Results in all regimes in ocean and atmosphere are displayed in Figure 4.9.Overall the tracer autocorrelations show shorter decorrelation time than the corresponding ad-vection flow modes in Figure 4.5. For the mid latitude case, the most important modes becomethe slow varying zonal modes, where the extremely long mixing time is still captured in themodel. Among all three test cases, the tracer autocorrelation functions can be captured withgood accuracy in the reduced-order models.

4.3.2. Prediction in tracer intermittency and fat-tails in PDFsAnother important issue in the reduced-order models is the prediction of intermittency in

the leading tracer modes. It has been observed from simulations in the true models that the trac-ers undergo a transition from Gaussian behavior to a probability distribution with approximatelyexponential fat-tails over a wide range of its variability with the change of model parameters. Inthe formulation in the tracer equations (2.4a), we can change the parameter value ε for scale sep-aration between the tracer field and turbulent flow. The change of values in ε leads to changesin the Peclet number Pe=UL/κ , where U is the characteristic scale in the advection flow v. Itis shown [3, 17, 22] that the passive tracer produces stronger intermittency as the tracer field ap-proaches the long time slow varying limit ε→0. Still here we would not like to push the systemto the extremes but consider two typical cases with ε =1/5 (where strong fat-tailed distributionsare developed) and ε =1 (where the tracer statistics become near Gaussian).

In Figure 4.10 and 4.11, we compare the high latitude atmosphere regime with two differentscale separation parameter value ε =1/5 and ε =1. First observe that in the true signals from


100

101

102

100


2>, upper layer truth

M1 = 10

M1 = 2

100

101

102

100


2>, lower layer

100

101

102

wavenumber

100

variance <Tk1

Tk2


2 4 6 8

100

2 4 6 8

100

2 4 6 8

100


100

101

102

100


k>, upper layer

100

101

102

100


k>, lower layer

100

101

102

wavenumber

100



2 4 6 8

10-2

100

2 4 6 8

100

2 4 6 810

-4

10-2

100


Fig. 4.7: Tracer variance spectra predicted with the reduced-order model in high latitude atmo-sphere regime with tracer parameters

(ε−1,dT

)=(5,0.1). The modes are ordered in descending

order of the corresponding advection flow energy. The truth is shown in black, and we considerdifferent number of resolved modes M1=10 and M1=2 in the reduced advection.

100

101

102

100


2>, upper layer

truth

red. model, M1 = 10

red. model, M1 = 2

100

101

102

100


2>, lower layer

100

101

102

wavenumber

100

variance spectra <Tk1

Tk2



100

101

102

10-5

100


k>, upper layer

truth

red. model, M1 = 10

red. model, M1 = 2

100

101

102

10-5

100

cross-covarinace spectra <Tkψ

k>, lower layer

100

101

102

wavenumber

10-5

100




Fig. 4.8: Tracer variance spectra predicted with the reduced-order model in mid latitude atmo-sphere regime with tracer parameters

(ε−1,dT

)=(5,0.1). The modes are ordered in descending

order of the corresponding advection flow energy. The truth is shown in black, and we considerdifferent number of resolved modes M1=10 and M1=2 in the reduced advection.


0 5 10 15 20 25 30 35 40-1

0

1mode (1,0) truth

red. model

0 20 40 60 80 100-1

0

1mode (0,1)

0 5 10 15 20 25 30-1

0

1mode (1,1)

0 5 10 15 20 25 30

time

-1

0

1mode (-1,1)

(a) tracer high lat ocean

0 10 20 30 40 50-1

0

1mode (1,0)

truth

red. model

0 20 40 60 80 100-1

0

1mode (0,1)

0 10 20 30 40 50-1

0

1mode (1,1)

0 10 20 30 40 50

time

-1

0

1mode (-1,1)

(b) tracer high lat atmosphere

0 50 100 150 200 250 300-1

0

1mode (0,1) truth

red. model

0 50 100 150 200 250 300-1

0

1mode (0,2)

0 20 40 60 80 100-1

0

1mode (1,1)

0 20 40 60 80 100

time

-1

0

1mode (-1,1)

(c) tracer mid lat atmosphere

Fig. 4.9: Prediction of the reduced-order autocorrelation functions in the tracer field in the firstfour most energetic modes. In the ocean case F =40, the tracer parameters are

(ε−1,dT

)=

(5,0.5); and in the atmosphere case F =4, the tracer parameters are(ε−1,dT

)=(5,0.1). In the

autocorrelation functions, the truth is shown in black dashed lines and the reduced-order modelprediction in red lines.

the perfect system with high resolution, larger spikes are produced in the tracer trajectories withsmaller value ε =1/5, and obvious fat tails can be observed in the leading modes representingstrong intermittent structures. On the other hand with larger value ε =1, the tracer field reducesto near Gaussian statistics where all the modes have the distributions close to the Gaussian fit.Similar case can be seen in the ocean regime case in Figure 4.12 and 4.13. Intermittency be-comes weaker as ε is in larger value. Both the near Gaussian and highly non-Gaussian regimescan be representative showing a wide range of variability in the passive tracer field.

Then we observe the reduced-order model skill in capturing these central non-Gaussianfat-tails. In Figure 4.10-4.13, we compare the imperfect model prediction in the high latituderegime in both ocean and atmosphere. Since the tracer intermittency is usually most importantin the first few leading modes, we compare the representative time-series and tracer PDFs instatistical steady state. Among all the regimes, the fat-tails in the distribution functions can becaptured, and similar characteristic structures can be seen in the truth and reduced model time-series. On the other hand in the nearly Gaussian regimes, still similar structures can be seen bycomparing the truth and imperfect model results, and the reduced-order model can recover thetracer PDFs again with accuracy.

Finally, in Figure 4.14 and 4.15, the mid latitude regime with persistent zonal jet is tested.In this case, the representative feature is the slow-varying leading zonal modes that becomedominant. In this case, the streamlines will be open with strong zonal flow for most of thetime in the general situation, thus the contour lines of the tracer mean gradient along y is lessfrequent to be broken. Thus tracer turbulent mixing in this case is weaker and fat-tails in thePDFs is less obvious. The reduced-order model maintains the skill in capturing tracer statisticsin leading modes among this regime. Notice here in all the reduced-order simulations, only thelargest scale modes |k|≤10 are resolved compared with the full model with resolution of 128grid points along each horizontal direction. The model efficiency can be effectively improvedby solving the low-order model.

Reduced model error due to truncated modes in tracer advection termIn the advection term vM ·∇TM in modeling the passive tracer field (2.7) in the reduced-

order approximation, only the resolved large-scale modes in the flow equation solution vM are


5 Concluding discussion 22

1000 1500 2000 2500 3000 3500 4000

-5

0

5time-series of mode (1,0) in lower layer, true model

1000 1500 2000 2500 3000 3500 4000

time

-5

0

5time-series of mode (1,0) in lower layer, reduced model

1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000-2

0


1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000

time

-2

0


-5 0 5

100

upper layer, mode (1,0)

-2 -1 0 1 2

100

mode (0,1)

-1.5 -1 -0.5 0 0.5 1 1.5

100

mode (1,1)

-1.5 -1 -0.5 0 0.5 1 1.5

100

mode (-1,1)

(c) upper layer

-5 0 5

100

lower layer, mode (1,0)

-2 -1 0 1 2

100

mode (0,1)

-2 -1 0 1 2

100

mode (1,1)

-2 -1 0 1 2

100

mode (-1,1)

(d) lower layer

Fig. 4.11: Prediction about tracer intermittency in high latitude atmosphere regime with parameterse1,dT

=

(5,0.1). The left panel is the time-series for the first two leading modes (1,0) and (0,1) between true modeland reduced model results; the right panel compares the PDFs in the first four modes between the truth andreduced model prediction.


1000 1500 2000 2500 3000 3500 4000

-5

0


1000 1500 2000 2500 3000 3500 4000

time

-5

0


1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000-2

0


1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000

time

-2

0


-5 0 5

100


-2 -1 0 1 2

100

mode (0,1)

-1.5 -1 -0.5 0 0.5 1 1.5

100

mode (1,1)

-1.5 -1 -0.5 0 0.5 1 1.5

100

mode (-1,1)

(c) upper layer

-5 0 5

100


-2 -1 0 1 2

100

mode (0,1)

-2 -1 0 1 2

100

mode (1,1)

-2 -1 0 1 2

100

mode (-1,1)

(d) lower layer


=

(5,0.1). The left panel is the time-series for the first two leading modes (1,0) and (0,1) between true modeland reduced model results; the right panel compares the PDFs in the first four modes between the truth andreduced model prediction.

Fig. 4.10: Prediction of tracer intermittency in high latitude atmosphere regime with parameters(ε−1,dT

)=(5,0.1). The left panel is the time-series for the first two leading modes (1,0) and

(0,1) between true model and reduced model results; the right panel compares the PDFs in thefirst four modes between the truth in blue and reduced model prediction in red with the Gaussianfit in dashed black lines.

used in calculating the nonlinear tracer advection. That is, the tracer advection in the model iscalculated through the large scale truncation. In above simulation results for tracer PDFs, wealways use the entire number of resolved flow modes M1=M from the advection velocity solu-tion vM . Here as a further test for model errors through the high wavenumber truncation, we caneven consider less modes M1<M in computing the nonlinear advection in the tracer dynamics.In Figure 4.16, PDFs in the two representative atmosphere and ocean regimes are comparedwith different truncation sizes M1=10,5,2 in computing the nonlinear tracer advection termwith vM. With M1=M=10, all the resolved modes from linear Gaussian flow model (2.6) areused in the tracer advection. Using half of the resolved modes, M1=5, still we observe that themajor structure with fat-tails can be captured in the leading PDFs with only a little larger errorin the tails of the distributions. In the extreme case using the severe truncation M1=2, only thefirst two modes are included in calculating the nonlinear advection. As a result, larger errorsappear with this crude approximation due to the insufficient quantification for flow advectionmodes.

4.3.3. Eddy diffusivity approximations for tracersParameterization by eddy diffusivity in the tracer system is often used in practice to account

for unresolved small scales [19, 17]. Here we demonstrate the accuracy of the imperfect modelin recovering the tracer eddy diffusivity. Consider the mean statistical dynamics about Tk=〈Tk〉of the tracer equations by taking ensemble average about the original model (2.4a)

dTk+(v·∇T )kdt+⟨v′ ·∇T ′

⟩kdt=Γkψkdt−(γT,k+iωT,k)Tkdt.

Above 〈v′ ·∇T ′〉k is the eddy diffusivity for the tracer dynamics. This is a spatially nonlocal terminvolving the interactions between all the spectral modes among all scales. In the turbulent-viscosity hypothesis [29, 19], the tracer turbulent advection is represented by a diffusion termas

⟨v′ ·∇T ′

⟩=−∇·(κT ∇T ), (4.3)



2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000-2

0


2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000

time

-2

0


2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000-2

0


2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000

time

-2

0


-2 -1.5 -1 -0.5 0 0.5 1 1.5 210-2

100


-1.5 -1 -0.5 0 0.5 1 1.5

100

mode (0,1)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 210-2

100


-1.5 -1 -0.5 0 0.5 1 1.5

100

mode (0,1)


=

(1,0.1). The left panel is the time-series for the first two leading modes (1,0) and (0,1) between true modeland reduced model results; the right panel compares the PDFs in the first four modes between the truth andreduced model prediction with the Gaussian fit in dashed black lines.


2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000-2

0


2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000

time

-2

0


2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000-2

0


2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000

time

-2

0


-2 -1.5 -1 -0.5 0 0.5 1 1.5 210-2

100


-1.5 -1 -0.5 0 0.5 1 1.5

100

mode (0,1)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 210-2

100


-1.5 -1 -0.5 0 0.5 1 1.5

100

mode (0,1)


=

(1,0.1). The left panel is the time-series for the first two leading modes (1,0) and (0,1) between true modeland reduced model results; the right panel compares the PDFs in the first four modes between the truth andreduced model prediction with the Gaussian fit in dashed black lines.

Fig. 4.11: Prediction of tracer intermittency in high latitude atmosphere regime with parameters(ε−1,dT



where the coefficient κT is a constant “eddy diffusivity”. The turbulent-viscosity hypothesisimplies that the scalar flux vector is aligned with the mean scalar gradient vector. It is introducedthrough a homogenization about the turbulent system.

Here we want to compare the eddy diffusivity from the truth with the reduced-order ap-proximations. From (4.3) we can calculate the eddy diffusivity coefficients for both the truesystem and reduced-order models through each spectral mode solution

κT =〈v′ ·∇T ′〉k|k|2 Tk

, κMT =〈v′M ·∇T ′M〉k|k|2 TM,k

. (4.4)

The statistics are achieved through time averaging along solution trajectories in both true systemand reduced stochastic model. Note that in the reduced-order models, only the first 10 leadingmodes are resolved, thus 〈v′M ·∇T ′M〉k are calculated based on these available resolved modesonly. Also, especially in homogeneous case, the statistical mean states of the tracer modesbecome small. Thus (4.4) can only qualitatively describe the mixing rate in each mode.

Figure 4.17 shows the results for eddy diffusivity estimated from (4.4) for mid latitude at-mosphere F =4 with jets. The modes are ordered in the same descending order according to theequilibrium statistical energy spectra of the vorticity as before. From the black lines for the truemodel eddy diffusivity, beginning from the intermediate scales from wavenumber around 10,the numerical estimation about eddy diffusivity κT stays in nearly constant values with smallfluctuations. This illustrates that turbulent-viscosity hypothesis is valid in tracer intermediate tosmall scales, and can actually offer good approximation in the eddy-diffusivity homogenizationas in (4.3). In the reduced-order predictions with model errors, eddy diffusivity approxima-tion κM

T has agreement in amplitude along the spectrum except the largest scales, showing thatthe most important fluctuation interactions in the reduced-order approximations are modeledwith accuracy. In the largest scale modes, larger errors appear and the eddy diffusivity approx-imation gets violated. This is related with the highly anisotropic structure in the largest scale



200 400 600 800 1000 1200 1400 1600

time

-10

0


200 400 600 800 1000 1200 1400 1600-10

0


200 400 600 800 1000 1200 1400 1600

time

-2

0

2

time-series of mode (0,1) in lower layer, reduced model200 400 600 800 1000 1200 1400 1600

-2

0

2

time-series of mode (0,1) in lower layer, true model

-5 0 5

100


-2 -1 0 1 2

100

mode (0,1)

-2 -1 0 1 2

100

mode (1,1)

-2 -1 0 1 2

100

mode (-1,1)

-5 0 5

100


-2 -1 0 1 2

100

mode (1,1)

-2 -1 0 1 2

100

mode (-1,1)

-2 -1 0 1 2

100

mode (0,1)

Fig. 4.13: Prediction about tracer intermittency in high latitude ocean regime with parameterse1,dT

= (5,0.5).

The left panel is the time-series for the first two leading modes (1,0) and (0,1) between true model andreduced model results; the right panel compares the PDFs in the first four modes between the truth andreduced model prediction with the Gaussian fit in dashed black lines.


200 400 600 800 1000 1200 1400 1600

time

-10

0


200 400 600 800 1000 1200 1400 1600-10

0


200 400 600 800 1000 1200 1400 1600

time

-2

0

2

time-series of mode (0,1) in lower layer, reduced model200 400 600 800 1000 1200 1400 1600

-2

0

2


-5 0 5

100


-2 -1 0 1 2

100

mode (0,1)

-2 -1 0 1 2

100

mode (1,1)

-2 -1 0 1 2

100

mode (-1,1)

-5 0 5

100


-2 -1 0 1 2

100

mode (1,1)

-2 -1 0 1 2

100

mode (-1,1)

-2 -1 0 1 2

100

mode (0,1)


= (5,0.5).


Fig. 4.12: Prediction of tracer intermittency in high latitude ocean regime with parameters(ε−1,dT



representing the zonal jets and the large errors in the approximation for the mean state with largefluctuations. The important practical issue of model error from eddy diffusivity approximationin the tracer statistics is an important research area worth further investigation in the future.



1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000

-2

0

2


1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000

time

-2

0

2

time-series of mode (1,0) in lower layer, reduced model

1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000-2

0


1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000

time

-2

0


-3 -2 -1 0 1 2 3

100


-1.5 -1 -0.5 0 0.5 1 1.5

100

mode (0,1)

-3 -2 -1 0 1 2 3

100


-1.5 -1 -0.5 0 0.5 1 1.5

100

mode (0,1)


= (1,0.5).


4 Numerical experiments for passive tracer statistics in various regimes 265 Concluding discussion 27

2000 3000 4000 5000 6000 7000 8000 9000 10000-0.5

0

0.5time-seris of mode (0,1) in tracer, true model

2000 3000 4000 5000 6000 7000 8000 9000 10000

time

-0.5

0

0.5time-seris of mode (0,1) in tracer, reduced model

2000 2500 3000 3500 4000 4500 5000-0.5

0


2000 2500 3000 3500 4000 4500 5000

time

-0.5

0


-0.2 -0.1 0 0.1 0.2

100


-0.2 -0.1 0 0.1 0.2

100

mode (0,2)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (1,1)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (-1,1)

-0.2 -0.1 0 0.1 0.2

100


-0.2 -0.1 0 0.1 0.2

100

mode (0,2)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (1,1)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (-1,1)

Fig. 4.16: Prediction about tracer intermittency in low latitude atmosphere regime with parameterse1,dT

=

(1,0.1). The left panel is the time-series for two leading modes (0,1) and (1,1) between true model andreduced model results; the right panel compares the PDFs in the first four modes between the truth andreduced model prediction with the Gaussian fit in dashed black lines.


2000 3000 4000 5000 6000 7000 8000 9000 10000-0.5

0


2000 3000 4000 5000 6000 7000 8000 9000 10000

time

-0.5

0


2000 2500 3000 3500 4000 4500 5000-0.5

0


2000 2500 3000 3500 4000 4500 5000

time

-0.5

0


-0.2 -0.1 0 0.1 0.2

100


-0.2 -0.1 0 0.1 0.2

100

mode (0,2)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (1,1)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (-1,1)

-0.2 -0.1 0 0.1 0.2

100


-0.2 -0.1 0 0.1 0.2

100

mode (0,2)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (1,1)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (-1,1)


=


Fig. 4.16: Prediction about tracer intermittency in mid latitude atmosphere regime with parameterse1,dT

=


-3 -2 -1 0 1 2 3

100


-1.5 -1 -0.5 0 0.5 1 1.5

100

mode (0,1)

-3 -2 -1 0 1 2 3

100


-1.5 -1 -0.5 0 0.5 1 1.5

100

mode (0,1)

Fig. 4.17Fig. 4.13: Prediction of tracer intermittency in high latitude ocean regime with parameters(ε−1,dT



5. Concluding discussionIn this paper we developed reduced-order stochastic models for predicting extreme events

and intermittency in passive scalar turbulence. The turbulent flow field is generated by a two-layer quasi-geostrophic model containing non-Gaussian and anistropic structures that can beobserved in many realistic natural flows. The passive tracer is transported by this advectionflow and forced by a mean tracer gradient. Various representative statistical features in thepassive tracer field can be generated through this system from near Gaussian distributions tohighly intermittent structures with fat-tailed PDFs in the leading tracer modes. This passivetracer turbulence advected by the complex high dimensional flow can be compared with themuch simpler model using the L-96 system as the advection flow investigated in [26], whichshares many similarities in statistical structures with the QG flow but gets much simpler dy-namical equations (in fact, the F =5 case in L-96 model has strong non-Gaussian statistics withleading dominant mode as the atmosphere regime with jets, and the F =8 case in L-96 modelcan be viewed as a simplification of the high latitude ocean with near-Gaussian homogeneousstatistics). Therefore the model reduction idea in [26] can be used as a guideline for the modeldevelopment here under more complex setup.

The complex strongly turbulent dynamical system requires proper reduced-order modelingstrategy of adopting simple advection flow models especially in situations when realistic highdimensional applications apply, but still maintaining the skill in recovering the most importantstatistical features in the tracer field. Linear Gaussian stochastic model is then introduced toapproximate the background flow field, and the tracer equations with advection terms using onlythe resolved large-scale modes are applied for efficient prediction. Calibration of the imperfectmodels with error is then required to achieve the optimal performance. First linear damping andrandom noise corrections are introduced in the reduced-order model to replace the expensivehigher order nonlinear interactions in the original flow equations. This is from a stochasticrealization of the statistical closure models in [21, 28]. High order feedbacks from small scales



1000 1500 2000 2500 3000 3500 4000 4500 5000-0.5

0

0.5time-series of mode (0,1) in tracer, true model

1000 1500 2000 2500 3000 3500 4000 4500 5000

time

-0.5

0


1000 1500 2000 2500 3000 3500 4000 4500 5000-1

0

1time-series of mode (1,1) in tracer, true model

1000 1500 2000 2500 3000 3500 4000 4500 5000

time

-1

0

1time-seris of mode (1,1) in tracer, reduced model

-0.4 -0.2 0 0.2 0.4

100


-0.3 -0.2 -0.1 0 0.1 0.2

100

mode (0,2)

-0.5 0 0.5

100

mode (-1,1)

-0.5 0 0.5

100

mode (1,1)

-0.4 -0.2 0 0.2 0.4

100


-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (0,2)

-0.5 0 0.5

100

mode (-1,1)

-0.5 0 0.5

100

mode (1,1)


=



1000 1500 2000 2500 3000 3500 4000 4500 5000-0.5

0

0.5time-series of mode (0,1) in tracer, true model

1000 1500 2000 2500 3000 3500 4000 4500 5000

time

-0.5

0


1000 1500 2000 2500 3000 3500 4000 4500 5000-1

0

1time-series of mode (1,1) in tracer, true model

1000 1500 2000 2500 3000 3500 4000 4500 5000

time

-1

0

1time-seris of mode (1,1) in tracer, reduced model

-0.4 -0.2 0 0.2 0.4

100


-0.3 -0.2 -0.1 0 0.1 0.2

100

mode (0,2)

-0.5 0 0.5

100

mode (-1,1)

-0.5 0 0.5

100

mode (1,1)

-0.4 -0.2 0 0.2 0.4

100


-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (0,2)

-0.5 0 0.5

100

mode (-1,1)

-0.5 0 0.5

100

mode (1,1)


=


Fig. 4.14: Prediction of tracer intermittency in mid latitude atmosphere regime with parameters(ε−1,dT

)=(5,0.1). The left panel is the time-series for two leading modes (0,1) and (1,1)

between true model and reduced model results; the right panel compares the PDFs in the firstfour modes between the truth in blue and reduced model prediction in red with the Gaussian fitin dashed black lines.

can be approximated by low order moments in equilibrium, thus the efficiency in calculatingthe imperfect model parameters is guaranteed. Second the model prediction skill is furtherimproved through a systematic framework by measuring the model autocorrelation functionsin the spectral representation with the information metric [26]. The time mixing scales in theleading modes can be modeled with accuracy. The skill of imperfect models in capturing thecrucial large-scale structures in tracer statistics is tested among a wide variety of dynamicalregimes ranging from strongly non-Gaussian to near Gaussian statistics with strong mixing. Thereduced-order stochastic model displays uniform skill in capturing both tracer energy spectra inequilibrium and fat-tails in the leading resolved modes, as well as a good approximation for thetracer eddy diffusivity in the atmosphere regime with jets. Besides, the skillful linear Gaussianstochastic modeling algorithm developed here should also be useful for other applications suchas accurate forecast of mean responses to changes in forcing [18] and efficient algorithms forstate estimation or data assimilation [10].

Acknowledgments. This research of the Andrew Majda is partially supported by the Of-fice of Naval Research through MURI N00014-16-1-2161 and DARPA through W911NF-15-1-0636. Di Qi is supported as a graduate research assistant on these grant.



2000 3000 4000 5000 6000 7000 8000 9000 10000-0.5

0


2000 3000 4000 5000 6000 7000 8000 9000 10000

time

-0.5

0


2000 2500 3000 3500 4000 4500 5000-0.5

0


2000 2500 3000 3500 4000 4500 5000

time

-0.5

0


-0.2 -0.1 0 0.1 0.2

100


-0.2 -0.1 0 0.1 0.2

100

mode (0,2)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (1,1)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (-1,1)

-0.2 -0.1 0 0.1 0.2

100


-0.2 -0.1 0 0.1 0.2

100

mode (0,2)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (1,1)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (-1,1)


=


4 Numerical experiments for passive tracer statistics in various regimes 27

-0.2 -0.1 0 0.1 0.2

100


-0.2 -0.1 0 0.1 0.2

100

mode (0,2)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (1,1)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (-1,1)

-0.2 -0.1 0 0.1 0.2

100


-0.2 -0.1 0 0.1 0.2

100

mode (0,2)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (1,1)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

100

mode (-1,1)

Fig. 4.18

the nonlinear tracer advection term with vM. With M1 = M = 10, all the resolved modes from linear Gaussian flowmodel (2.5) are used in the tracer advection. Using half of the resolved modes, M1 = 5, still we observe that the majorstructure with fat-tails can be captured in the leading PDFs with only a little larger error in the tails of the distributions.In the extreme case using the severe truncation M1 = 2, only the first 4 modes are included in the nonlinear advectionterm. As a result, larger errors appear with this crude approximation due to the insufficient quantification for flowadvection modes.

4.3.3 Eddy diffusivity approximations for tracers

In the final part of the test for reduced-order models, we check the actual form of tracer eddy diffusivity [24, 21] and theimperfect models’ skill in recovering it. Consider the mean statistical dynamics about Tk = hTki of the tracer equationsby taking ensemble average about the original model (2.3a)

dTk +(v ·—T )k dt +v0 ·—T 0↵

k dt = Gkykdt (gT,k + iwT,k) Tkdt.

On the left hand side of the above equation, hv0 ·—T 0ik is the eddy diffusivity for the tracer dynamics. This is a spatiallynonlocal term involving the interactions between all the spectral modes among all scales. In the turbulent-viscosityhypothesis [34, 24], the tracer turbulent advection is represented by a diffusion term as

v0 ·—T 0↵= — · (kT —T ) , (4.3)

where the coefficient kT is a constant “eddy diffusivity”. The turbulent-viscosity hypothesis implies that the scalar fluxvector is aligned with the mean scalar gradient vector. It is introduced through a homogenization about the turbulentsystem.

Here we want to compare the eddy diffusivity from the truth with the reduced-order approximations. From (4.3) wecan calculate the eddy diffusivity coefficient for both the true system and reduced-order models through each spectralmode solution

kT =hv0 ·—T 0ik

|k|2 Tk, kM

T =hv0M ·—T 0

Mik

|k|2 TM,k. (4.4)

Fig. 4.15: Prediction of tracer intermittency in mid latitude atmosphere regime with parameters(ε−1,dT

)=(1,0.1). The left panel is the time-series for two leading modes (0,1) and (1,1)

between true model and reduced model results; the right panel compares the PDFs in the firstfour modes between the truth in blue and reduced model prediction in red with the Gaussian fitin dashed black lines.

-6 -4 -2 0 2 4 6

10-2

100

mode (1,0)

truth

M1 = 10

M1 = 5

M1 = 2

-1.5 -1 -0.5 0 0.5 1 1.510

-4

10-2

100

mode (0,1)

(a) high lat. atmosphere

-6 -4 -2 0 2 4 610

-4

10-2

100

mode (1,0)

truth

M1 = 10

M1 = 5

M1 = 2

-1.5 -1 -0.5 0 0.5 1 1.510

-4

10-2

100

mode (0,1)

(b) high lat. ocean

Fig. 4.16: Tracer PDFs in the first two leading modes (1,0) and (0,1) with different truncationM1=10,5,2 in computing the advection velocity vM . The truth in black is compared with thereduced-order model predictions with different reduction in the nonlinear advection term.

Appendix A. Information criterion for measuring autocorrelation functions of the sta-tionary random fields.

A.1. Spectral representation of general stationary random fieldsFor a scalar stationary process u(t), its autocorrelation function R (t)=

⟨u(0)u(t)∗

⟩is


100

101

102

100

eddy diffusivity, upper layer

truth

red. model

100

101

102

wavenumber

100

eddy diffusivity, lower layer

(a)(ε−1,dT

)=(5,0.1)

100

101

102

100

eddy diffusivity, upper layer

truth

red. model

100

101

102

wavenumber

100

eddy diffusivity, lower layer

(b)(ε−1,dT

)=(1,0.1)

Fig. 4.17: Eddy diffusivity calculated from (4.4) for the mid latitude atmosphere regime withdifferent parameters. The modes are ordered according to the descending order of the advectionflow vorticity variance. The true model results are shown in dashed black lines, while thereduced-order model predictions are in red.

positive-definite in time, that is,

N

∑i, j=1

R (ti−t j)aia∗j≥0, ∀a1,···aN∈CN .

Thus it is proved by Khinchin’s formula [35] that the autocorrelation function has a spectralrepresentation

R (t)=ˆ ∞

−∞eiλ tdF (λ )=

ˆ ∞

−∞eiλ tE (λ )dλ , (A.1)

where F (λ ) is the non-decreasing spectral distribution function of the stationary process u(t),and E (λ )>0 is the spectral density function of the stationary process u(t). According to theformula (A.1), the spectral distribution function F (λ ) of the process u(t) can be determinedfrom its autocorrelation function R (t) via Fourier transform. Conversely, the autocorrelationfunction can be calculated from the corresponding spectral distribution function. Therefore wecan construct the spectral representation of the stationary random field u(t) as

u(t)=ˆ ∞

−∞eiλ t Z(dλ ), (A.2)

with Z(dλ ) a random measure which can be defined in a similar form of Fourier transform as

Z(dλ )= limT→∞

dλ2T

ˆ T

−Te−iλ tu(t)dt.

By combining (A.1) and (A.2) we can find that E (λ ) or dF (λ ) can be viewed as the energyspectrum of the random field u which measures the variance in Z(dλ )

dF (λ )=E (λ )dλ =E∣∣Z(dλ )

∣∣2 . (A.3)

Note here that in general Z(dλ ), which depends on the statistics of the stationary process u(t), isnot necessarily a Gaussian random variable and may include higher order statistical information.But in the main text we only concern fitting a linear Gaussian process with consistent statistics


up to second order, so a simple Gaussian approximation uG(t) with statistics up to secondorder moments is sufficient. The true model autocorrelation function R (t) and correspondingspectral density E (λ ) can be achieved directly through the Fourier transform (A.1) based ontrue observation of the data. As the discussion in the main text, we use an Ornstein-Uhlenbeckprocess to approximate the flow model, that is,

duM =−(γ+iω)uMdt+σdW.

The autocorrelation function and corresponding spectral density function can be calculated inexact forms,

RM (t)=exp(−(γ+iω)t), EM (λ )=ˆ ∞

−∞e−iλ tR (t)dt=

2γγ2+(λ +ω)2 . (A.4)

A.2. Spectral information criterion for measuring the autocorrelation functions intwo-layer model

In the linear stochastic model (2.6) of advection flow vorticity~qM (t), one additional differ-ence is that we are considering a 2-vector field. Still this can be easily reduced to the previousscalar case. In fact in the stochastic approximation model, (2.6) becomes a two-dimensionalOrnstein-Uhlenbeck process for each vorticity spectral mode ~qM =(q1,q2)

T including the up-per and lower layer modes

d~qM =−(Γ+iΩ)~qM+Σd~W ,

where Γ,Ω,Σ are 2×2 matrices and we neglect the subscripts in the state variables. Explicitlyin the two-layer QG model we have

Γ+iΩ=

[0

r|k|2]

H−1k +ν |k|2s I2+DM

q,k+ikx

[U−U

]+ikx

[β +k2

dUβ−k2

dU

]H−1

k ,Σ=ΣMq,k,

with the original linear operators (2.6) and additional imperfect model corrections (3.6) com-bined together. The autocorrelation function about the above system becomes the 2×2 matrix

RM(t,t ′)≡⟨~qM (t)~qM

(t ′)∗⟩

=

ˆ t∧t ′

t0e−(t−s)(Γ+iΩ)Σ2e−(Γ−iΩ)(t ′−s)ds. (A.5)

In each entry of the above matrix, the integrant is still an exponential about time; then in sta-tionary state the autocorrelation function is only dependent on the time difference |t ′−t|. Thuseach entry of the above autocorrelation matrix can be reduced to the exact form of the scalarcase (A.4). The corresponding spectral density EM (λ ) is the 2×2 matrix that can be definedcomponentwisely as in (A.4).

The true process constraint in Gaussian statistics, ~qG, and the imperfect linear modelapproximation, ~qM (t), can be then naturally expressed under an independent 2-dimensionalWiener random measure d ~W (λ )=

(dW1,dW2

). That is,

~qG(t)=ˆ ∞

−∞eiλ tE1/2(λ )d ~W (λ ), (A.6)

~qM (t)=ˆ ∞

−∞eiλ tE

1/2M (λ )d ~WM (λ ). (A.7)

Above E (λ ) is the true spectral energy, while EM (λ ) is the explicit form for the density ofautocorrelation function for the linear model from (A.5). Instead of measuring the information


distance between the stationary random fields~qG(t) and~qM (t) at fixed time t, a more favorableway is to compare the distributions between the true spectral random measure and the imperfectmodel approximation field which can include the information about the autocorrelation func-tions. With the help of the spectral decomposition of stationary random process, we decomposethe state variables into the spectral forms in (A.6) and (A.7) respectively under each Fourier ba-sis

eiλ t

in time and measure the information distance between one single spectral incrementat frequency λ . Here consider the full spectra of random measures

pG=∏λ

pG(x;λ )∼

E1/2(λ )d ~W (λ )∞

λ=−∞

and

pM =∏λ

pM (x;λ )∼

E1/2M (λ )d ~WM (λ )

∞

λ=−∞

as two random process about λ ∈R. Due to the independent increments between ~W (dλ1) and~W (dλ2), the information distance between these two random processes can be expressed as theintegration between all the spectral modes [26, 35]

P (pG,pM)=P

(∏λ

pG(x;λ ),∏λ

pM (x;λ )

)

=

ˆ ∞

−∞dλP (pG(x;λ ),pM (x;λ ))

=

ˆdλD

(E (λ )E−1

M (λ )),

(A.8)

where D (x)=−logdetx+trx−N is the Gaussian relative entropy with a zero mean state.

REFERENCES

[1] M. AVELLANEDA AND A. J. MAJDA, Simple examples with features of renormalization for turbulent transport,Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and EngineeringSciences, 346 (1994), pp. 205–233.

[2] P. BARTELLO AND G. HOLLOWAY, Passive scalar transport in β -plane turbulence, Journal of Fluid Mechanics,223 (1991), pp. 521–536.

[3] A. BOURLIOUX AND A. MAJDA, Elementary models with probability distribution function intermittency forpassive scalars with a mean gradient, Physics of Fluids (1994-present), 14 (2002), pp. 881–897.

[4] J. C. BRONSKI AND R. M. MCLAUGHLIN, The problem of moments and the Majda model for scalar intermit-tency, Physics Letters A, 265 (2000), pp. 257–263.

[5] R. CAMASSA, N. MARTINSEN-BURRELL, AND R. M. MCLAUGHLIN, Dynamics of probability density func-tions for decaying passive scalars in periodic velocity fields, Physics of Fluids, 19 (2007), p. 117104.

[6] T. DELSOLE, Stochastic models of quasigeostrophic turbulence, Surveys in Geophysics, 25 (2004), pp. 107–149.[7] T. DELSOLE AND B. F. FARRELL, A stochastically excited linear system as a model for quasigeostrophic

turbulence: Analytic results for one-and two-layer fluids, Journal of the atmospheric sciences, 52 (1995),pp. 2531–2547.

[8] G. FALKOVICH, K. GAWEDZKI, AND M. VERGASSOLA, Particles and fields in fluid turbulence, Reviews ofmodern Physics, 73 (2001), p. 913.

[9] D. M. FRIERSON, Robust increases in midlatitude static stability in simulations of global warming, Geophysicalresearch letters, 33 (2006).

[10] B. GERSHGORIN AND A. J. MAJDA, Filtering a statistically exactly solvable test model for turbulent tracersfrom partial observations, Journal of Computational Physics, 230 (2011), pp. 1602–1638.

[11] J. GOULD, D. ROEMMICH, S. WIJFFELS, H. FREELAND, M. IGNASZEWSKY, X. JIANPING, S. POULIQUEN,Y. DESAUBIES, U. SEND, K. RADHAKRISHNAN, ET AL., Argo profiling floats bring new era of in situocean observations, Eos, 85 (2004), pp. 179–184.


[12] J. HARLIM, A. MAHDI, AND A. J. MAJDA, An ensemble kalman filter for statistical estimation of physicsconstrained nonlinear regression models, Journal of Computational Physics, 257 (2014), pp. 782–812.

[13] R. KLEEMAN AND A. MAJDA, Predictability in a model of geophysical turbulence, Journal of the atmosphericsciences, 62 (2005), pp. 2864–2879.

[14] S. KULLBACK AND R. A. LEIBLER, On information and sufficiency, The annals of mathematical statistics, 22(1951), pp. 79–86.

[15] Y. LEE, A. J. MAJDA, AND D. QI, Stochastic superparameterization and multiscale filtering of turbulent trac-ers, Multiscale Modeling & Simulation, 15 (2017), pp. 215–234.

[16] A. MAJDA, R. V. ABRAMOV, AND M. J. GROTE, Information theory and stochastics for multiscale nonlinearsystems, vol. 25, American Mathematical Soc., 2005.

[17] A. J. MAJDA AND B. GERSHGORIN, Elementary models for turbulent diffusion with complex physical features:eddy diffusivity, spectrum and intermittency, Phil. Trans. R. Soc. A, 371 (2013), p. 2012.0184.

[18] A. J. MAJDA, B. GERSHGORIN, AND Y. YUAN, Low-frequency climate response and fluctuation-dissipationtheorems: theory and practice, Journal of the Atmospheric Sciences, 67 (2010), pp. 1186–1201.

[19] A. J. MAJDA AND P. R. KRAMER, Simplified models for turbulent diffusion: theory, numerical modelling, andphysical phenomena, Physics reports, 314 (1999), pp. 237–574.

[20] A. J. MAJDA AND D. QI, Improving prediction skill of imperfect turbulent models through statistical responseand information theory, Journal of Nonlinear Science, 26 (2016), pp. 233–285.

[21] , Strategies for reduced-order models for predicting the statistical responses and uncertainty quantifica-tion in complex turbulent dynamical systems, submitted to SIAM Review, (2016).

[22] A. J. MAJDA AND X. T. TONG, Intermittency in turbulent diffusion models with a mean gradient, Nonlinearity,28 (2015), pp. 4171–4208.

[23] M. A. MOHAMAD AND T. P. SAPSIS, Probabilistic description of extreme events in intermittently unstabledynamical systems excited by correlated stochastic processes, SIAM/ASA Journal on Uncertainty Quantifi-cation, 3 (2015), pp. 709–736.

[24] J. D. NEELIN, B. R. LINTNER, B. TIAN, Q. LI, L. ZHANG, P. K. PATRA, M. T. CHAHINE, AND S. N.STECHMANN, Long tails in deep columns of natural and anthropogenic tropospheric tracers, GeophysicalResearch Letters, 37 (2010).

[25] S. B. POPE, Turbulent flows, IOP Publishing, Bristol, 2001.[26] D. QI AND A. J. MAJDA, Predicting fat-tailed intermittent probability distributions in passive scalar turbulence

with imperfect models through empirical information theory, Comm. Math. Sci, 14 (2015), pp. 1687–1722.[27] , Low-dimensional reduced-order models for statistical response and uncertainty quantification:

barotropic turbulence with topography, Physica D: Nonlinear Phenomena, (2016).[28] , Low-dimensional reduced-order models for statistical response and uncertainty quantification: two-

layer baroclinic turbulence, Journal of the Atmospheric Sciences, 73 (2016), pp. 4609–4639.[29] R. SALMON, Lectures on geophysical fluid dynamics, Oxford University Press, 1998.[30] T. P. SAPSIS AND A. J. MAJDA, Statistically accurate low-order models for uncertainty quantification in turbu-

lent dynamical systems, Proceedings of the National Academy of Sciences, 110 (2013), pp. 13705–13710.[31] K. S. SMITH, Tracer transport along and across coherent jets in two-dimensional turbulent flow, Journal of

Fluid Mechanics, 544 (2005), pp. 133–142.[32] A. F. THOMPSON AND W. R. YOUNG, Two-layer baroclinic eddy heat fluxes: Zonal flows and energy balance,

Journal of the Atmospheric Sciences, 64 (2007), pp. 3214–3231.[33] G. K. VALLIS, Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation, Cambridge

University Press, 2006.[34] E. VANDEN EIJNDEN, Non-gaussian invariant measures for the Majda model of decaying turbulent transport,

Communications on Pure and Applied Mathematics, 54 (2001), pp. 1146–1167.[35] A. M. YAGLOM, An introduction to the theory of stationary random functions, Courier Corporation, 2004.[36] P. ZURITA-GOTOR, J. BLANCO-FUENTES, AND E. P. GERBER, The impact of baroclinic eddy feedback on

the persistence of jet variability in the two-layer model, Journal of the Atmospheric Sciences, 71 (2014),pp. 410–429.

Documents

Courant Institute of Mathematical Sciences - …qidi/publications/tracer_twolayer.pdfPassive scalar turbulence concerns about the advection and diffusion of a tracer ﬁeld pas-sively